There's no other reservoir. The water down the overflow riser goes to exactly the same place that the water going past the flapper goes -- the bowl.
I started to tell you that disconnecting the refill hose will make zero difference whatsoever if it's daylighted above the top of the overflow riser and thus above the water level in the tank. That's certainly the case if the source of the problem is the tank. In that case, most likely you would have either have a crack in the overflow riser or a flapper past which water is leaking.
The typical test for that is to turn the water off at the wall, and let the thing sit overnight. Mark the level in the tank when you start and see where it is in the morning. If the water has dropped to a specific level, look for a crack in the riser there (or a vertical crack that ends there; pinch the riser to see if it separates). If the tank is empty, there is likely just a bad seal near the flapper.
However, the other possibility is that your bowl is overfilling, and you're hearing the water slowly trickle over the weir as the bowl level settles. Mark the water level in the bowl immediately after your fill valve shuts off, and then wait an hour and see if it has trickled lower. Actually, now that I think about it, that WOULD implicate disconnecting the refill tube because you would have less bowl refill in that event. Indeed, I'm guessing that you will find that this is the issue.
If so, then it's possible that the fill valve adjustment (if it's an adjustable one) just needs to be ratcheted back a little bit. If not, and this is the second one, I have to wonder if your toilet is level front-to-back. If it's tilted back a bit, the bowl won't hold as much water before it flows over the weir and while the correct amount is flowing into the bowl on refill, some is flowing over the weir because of a slight tilt. Regardless, it's an easy fix. Let us know.
